I'm going out on a limb here, to see if anyone can help me or point me into the correct direction:

The problem is simple, but very hard to answer as we make some of the numbers bigger. Lets start of with a simple example:

In this problem, I have Columns, that are hollow, like pipes, and they have space in them for 2 balls. I can represent them with a 2 x 2 matrix when empty,

[ 0 , 0 ][ 0 , 0 ]

and full

[ 1 , 1 ][ 1 , 1 ]

The problem comes, is determining the probability of "filling" up a column or tube when a ball is placed randomly, The only information that we keep after each ball is placed in, is how many we have previously placed.

This 2 x 2 example is incredibly easy. Here it is as follows, Let f(N) = the probability that a ball will fill a column, where N is the number of balls previously in the array.